
where \(\gamma =r^2/A\)  is the determinant of the spatial two-metric obtained from (REF ) by setting \(t\)  to be constant. We used the boundary conditions on the function \(a(r)\) : \(a(R)=n\)  and \(a(0)=0\) . Note that the expression for the magnetic flux \(\Phi =\frac{2\,\pi \,n}{e}\)  is the same in curved space as it is in fixed Minkowski spacetime [1]}. In the next section where we present our numerical results, we will integrate numerically over the area the different magnetic field profiles for different coupling \(\xi \)  and show that the result is the same independent of the profile and \(\xi \) . Besides demonstrating numerically that the magnetic flux is a topological invariant in curved space, it also provides another check on our numerical simulation. The magnetic flux is “quantized" as it comes in integer steps of \(2\pi /e\) . This does not stem from any quantization procedure imposed on the fields but from the topology of the vortex which is characterized by its winding number \(n\) .
